Search

In this note it is established that a finite family of positive linear operators acting from an Archimedean vector lattice into an Archimedean -algebra with unit is disjointness preserving if and only if the polynomial presented in the form of the product of powers of these operators is orthogonally additive. A similar statement is established for the sum of polynomials represented as products of powers of positive operators. [ABSTRACT FROM AUTHOR]

We establish the existence of a simultaneous extension from a majorizing sublattice in the classes of regular multilinear operators and regular homogeneous polynomials on vector lattices. By simultaneous extension from a sublattice we mean a right inverse of the restriction operator to this sublattice which is an order continuous lattice homomorphism. The main theorems generalize some earlier results for orthogonally additive polynomials and bilinear operators. The proofs base on linearization by Fremlin's tensor product and the existence of a right inverse of an order continuous operator with Levy and Maharam property. [ABSTRACT FROM AUTHOR]

Using the linearization of positive multilinear operators by means of the Fremlin tensor product of vector lattices makes it possible to show that a multilinear operator from the Cartesian product of majorizing subspaces of vector lattices to Dedekind complete vector lattice admits extension to a positive multilinear operator on the Cartesian product of the ambient vector lattices. We establish that this is valid if the multilinear operator is defined on the Cartesian product of majorizing subspaces of separable Banach lattices and takes values in a topological vector lattice with the -interpolation property, provided that the Banach lattices have the property of subadditivity. The latter ensures that the algebraic tensor product of the majorizing subspaces is majorizing in the Fremlin tensor product of the Banach lattices. The open question is: whether or not the result is valid if the subadditivity property is omitted or weakened. The possibility of weakening the order completeness of the target lattice by some additional requirements on the initial vector lattices was firstly observed by Abramovich and Wickstead in proving a version of the Hahn–Banach–Kantorovich theorem. [ABSTRACT FROM AUTHOR]

A lattice homomorphism between quasi-Banach lattices is known to be compact if and only if it is a sum of a series of rank one lattice homomorphisms converging in the operator norm with pairwise disjoint images. We obtain an analogous description for the dual class of -compact and compact linear operators that almost preserve intervals and act in quasi-Banach lattices. As a corollary, we get a characterization of a pair of quasi-Banach lattices having no nonzero -compact (compact) operators that almost preserve intervals. Also, we prove some theorems of the Radon–Nikodym type for almost interval preserving -compact (compact) operators. [ABSTRACT FROM AUTHOR]

The main result states that if E is a separable Fréchet lattice and F is a (locally solid) topological vector lattice with the σ-interpolation property then each positive linear operator T0 from a majorizing subspace G ⊂ E into F admits extension to a continuous positive linear operator T from E into F. This fact is proved by using only the axiom of countable choice. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions are found under which the sum of N order bounded disjointness preserving operators is n-disjoint with n and N naturals. It is shown that the decomposition of an order bounded n-disjoint operator into a sum of disjointness preserving operators is unique up to "Boolean permutation," the meaning of which is clarified in the course of the presentation. [ABSTRACT FROM AUTHOR]

We prove an analog of the Dodds-Fremlin-Wickstead Theorem on compact domination for homogeneous orthogonally additive polynomials in Banach lattices. The proof is based on linearization of the polynomials which was established earlier by the author. [ABSTRACT FROM AUTHOR]

It is proved that an orthogonally additive order bounded homogeneous polynomial acting between uniformly complete vector lattices admits a representation in the form of the composition of a linear order bounded operator and a special homogeneous polynomial playing the role of a power-law function, which is absent in the vector lattice. This result helps to establish a criterion for the integral representability of an orthogonally additive homogeneous polynomial. [ABSTRACT FROM AUTHOR]

We prove that each bounded orthogonally additive homogeneous polynomial acting from an Archimedean vector lattice into a separated convex bornological space, under the additional assumption that the bornological space is complete or the vector lattice is uniformly complete, can be represented as the composite of a bounded linear operator and a special homogeneous polynomial which plays the role of the exponentiation absent in the vector lattice. The approach suggested is based on the notions of convex bornology and vector lattice power. [ABSTRACT FROM AUTHOR]